NTA JEE Mains 24th jan 2023 Shift 1

Let $$p, q \in \mathbb{R}$$ and $$(1 - \sqrt{3}i)^{200} = 2^{199}(p + iq)$$, $$i = \sqrt{-1}$$. Then, $$p + q + q^2$$ and $$p - q + q^2$$ are roots of the equation.

For three positive integers $$p, q, r$$, $$x^{pq^2} = y^{qr} = z^{p^2r}$$ and $$r = pq + 1$$ such that $$3, 3\log_y x, 3\log_z y, 7\log_x z$$ are in A.P. with common difference $$\frac{1}{2}$$. The $$r - p - q$$ is equal to

The value of $$\sum_{r=0}^{22} {^{22}C_r} \cdot {^{23}C_r}$$ is

Let a tangent to the curve $$y^2 = 24x$$ meet the curve $$xy = 2$$ at the points $$A$$ and $$B$$. Then the midpoints of such line segments $$AB$$ lie on a parabola with the

$$\lim_{t \to 0} \left(1^{\frac{1}{\sin^2 t}} + 2^{\frac{1}{\sin^2 t}} + 3^{\frac{1}{\sin^2 t}} \cdots n^{\frac{1}{\sin^2 t}}\right)^{\sin^2 t}$$ is equal to

The compound statement $$(\sim(P \wedge Q)) \vee ((\sim P) \wedge Q) \Rightarrow ((\sim P) \wedge (\sim Q))$$ is equivalent to

The relation $$R = \{(a, b) : gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\}$$ is:

If $$A$$ and $$B$$ are two non-zero $$n \times n$$ matrices such that $$A^2 + B = A^2B$$, then

Let $$N$$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$$x + y + z = 1$$, $$2x + Ny + 2z = 2$$, $$3x + 3y + Nz = 3$$
has unique solution is $$\frac{k}{6}$$, then the sum of value of $$k$$ and all possible values of $$N$$ is

Let $$\alpha$$ be a root of the equation $$(a-c)x^2 + (b-a)x + (c-b) = 0$$ where $$a, b, c$$ are distinct real numbers such that the matrix $$\begin{pmatrix} \alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c \end{pmatrix}$$ is singular. Then the value of $$\frac{(a-c)^2}{(b-a)(c-b)} + \frac{(b-a)^2}{(a-c)(c-b)} + \frac{(c-b)^2}{(a-c)(b-a)}$$ is